Dipole in a Uniform External Field

The magnitude of an electric dipole moment is p = qd, where q is the magnitude of either charge and d is the separation between the positive and negative charges. If q remains constant, p is directly proportional to d. Increasing the separation therefore increases the dipole moment in the same ratio. The field direction, object colour, and time do not appear in this definition. Thus option A is correct.

Torque is the turning effect produced by a force. For an electric dipole in a uniform field, the torque tends to rotate the dipole so that its dipole-moment vector changes direction relative to the field. It does not alter the magnitudes of the positive and negative charges, create the external field, or change the identity of the dipole. Therefore, option B is correct.

For a dipole in a uniform electric field, torque is τ = pE sinθ, where θ is the angle between the dipole moment and the field. Parallel alignment means θ = 0°, and sin0° = 0. Therefore τ = 4 × 3 × 0 = 0 N m. The nonzero product pE alone is not the torque; the sine factor is essential. Option C is correct.

The torque magnitude is τ = pE sinθ. Its maximum value occurs when sinθ = 1, which means the dipole is perpendicular to the electric field at θ = 90°. Therefore τmax = pE = 8 × 5 = 40 N m. The other choices use one input, add the inputs, or omit the product, so option D is the only correct value.

The magnitude of dipole moment is p = qd, where q is the magnitude of either charge and d is the separation. If q changes to 2q and d changes to d/2, the new moment is p′ = (2q)(d/2) = qd = p. Therefore the dipole moment remains unchanged. Options A, B, and D do not match the product of the two scale factors.

Dipole moment magnitude is defined by p = qd. If the charge becomes 3q and the separation becomes 2d, the new value is p′ = (3q)(2d) = 6qd = 6p. Both changes multiply the original moment, so their factors must be multiplied rather than added. Hence the dipole moment becomes six times and option D is correct.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ. Substituting p = 3 C m, E = 6 N/C, and θ = 90°, we get τ = 3 × 6 × sin 90° = 18 × 1 = 18 N m. Therefore option B is correct. Option A results from an incorrect multiplication, while C and D ignore the full product or the fact that sin 90° equals one.

The torque on an electric dipole is τ = pE sin θ, where θ is the angle between the dipole moment and the electric field. For a parallel orientation, θ = 0° and sin 0° = 0. Hence τ = 8 × 2 × 0 = 0 N m, so option C is correct. The product pE would be 16 N m only for maximum torque, not for the parallel position.

Using the governing equation τ = pE sin θ, substitute p = 5 C m, E = 4 N/C, and sin 30° = 1/2. Thus τ = 5 × 4 × 1/2 = 10 N m. Option A is correct. The value 20 N m in option B is pE without the angular factor, while C and D do not follow the required multiplication and sine dependence.

The torque magnitude is τ = pE sin θ. Its maximum value occurs when sin θ = 1, which means θ = 90°. Therefore τmax = pE = 6 × 8 = 48 N m. Option B is correct. The value 14 is an addition, whereas options C and D merely repeat one of the supplied quantities and do not apply the maximum-torque relation.

For a dipole, the maximum torque is τmax = pE because sin 90° = 1. Rearranging gives p = τmax/E. With τmax = 30 N m and E = 5 N/C, p = 30/5 = 6 C m. Hence option A is correct. Option B confuses the field with the dipole moment, option C adds the values, and option D multiplies them instead of dividing.

At maximum torque, τmax = pE because the dipole is perpendicular to the field and sin 90° = 1. Solving for the field gives E = τmax/p = 36/(9) = 4 N/C. Thus option A is correct. Option B uses p itself, while options C and D arise from subtraction or addition and do not satisfy the governing equation or units.

For an electric dipole, U = −pE cos θ. At the antiparallel position, θ = 180° and U = +pE, which is the maximum potential energy. At the parallel position, θ = 0° and U = −pE, the minimum value. When released, the electric torque drives the dipole toward the stable parallel orientation, so its potential energy decreases. Option A is correct; it does not increase or remain constant during the rotation.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. For θ = 60°, cos 60° = 1/2. Substitution gives U = −pE(1/2) = −pE/2. Therefore option A is correct. The positive expression ignores the negative sign in the energy formula, pE corresponds to the antiparallel orientation, and zero applies only when the dipole is perpendicular to the field.

Use the dipole-energy equation U = −pE cos θ. Here pE = (6)(2) = 12 J and cos 60° = 1/2. Thus U = −12 × 1/2 = −6 J. Option B is therefore correct. A positive 6 J results from missing the negative sign, 12 J would correspond to using cos θ = 1, and zero energy would require θ = 90°, not 60°.

The torque on an electric dipole is τ = pE sin θ, while the maximum torque is τ_max = pE, reached at 90°. Hence τ/τ_max = sin θ. For θ = 30°, sin 30° = 1/2, so the torque is one-half of its maximum value. Option A is correct. √3/2 belongs to 60°, unity is the maximum itself, and zero occurs at 0° or 180°.

For a dipole, τ = pE sin θ and τ_max = pE. Dividing these equations gives τ/τ_max = sin θ. At θ = 60°, sin 60° = √3/2, so the actual torque is √3/2 of the maximum torque. Therefore option B is correct. One-half is the value for 30°, one-fourth is not the required sine value, and zero applies at 0° or 180°.

The dipole torque is τ = pE sin θ and is maximum when θ = 90°. Its potential energy is U = −pE cos θ. Since cos 90° = 0, the potential energy at the maximum-torque orientation is U = 0, taking the usual zero reference used for this expression. Therefore option C is correct. Minimum and maximum energies occur at parallel and antiparallel orientations, where the torque is zero.

The governing relation is τ = pE sin θ, where p is the dipole moment and θ is the angle between the dipole moment and the electric field. Here p = 4 C m, E = 5 N/C, and sin 90° = 1. Thus τ = 4 × 5 × 1 = 20 N m. Option B would result from an incorrect operation, while option D applies only when the angle is 0° or 180°.

For a dipole in a uniform electric field, the torque is τ = pE sin θ. In the parallel position, the angle between the dipole moment and the field is θ = 0°, so sin 0° = 0. Hence τ = 7 × 3 × 0 = 0 N m. The product 21 N m would be the maximum value only if the angle were 90°, not when the dipole is parallel.

The torque on a dipole is τ = pE sin θ. Its maximum value occurs when sin θ = 1, that is, at θ = 90°. Therefore τmax = pE = 5 × 8 = 40 N m. The values 13, 8, and 5 arise from addition or using only one given quantity; none includes the correct product pE at the maximizing angle.

The governing expression is U = −pE cos θ. For the perpendicular position, θ = 90° and cos 90° = 0. Consequently, U = −(10)(3)(0) = 0 J. The values +30 J and −30 J correspond to the opposite and parallel orientations, respectively, while 10 J does not follow from the energy relation. Thus option C is the only consistent answer.

The governing relations are torque τ = pE sin θ and potential energy U = −pE cos θ, where p points from the negative charge to the positive charge. Thus, first establish the dipole direction and the angle with the field, then select the required relation and substitute values with units. Option A is correct; the other choices ignore the physical setup or use no valid solving method.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ. For θ = 90°, cos 90° = 0, so U = −pE × 0 = 0. Therefore option C is correct. It is not the maximum positive or negative value: those occur at 180° and 0°, respectively. The result assumes the usual reference for dipole potential energy in a uniform field.

The magnitude of torque on an electric dipole in a uniform field is τ = pE sin θ. If the dipole moment p and angle θ remain unchanged, torque is directly proportional to the field E. Replacing E by 2E gives τ′ = p(2E)sinθ = 2τ. Therefore, the torque doubles; it does not become half, four times, or zero.